On local Tur\'an density problems of hypergraphs

TL;DR

This paper investigates the local Turán density problem in hypergraphs, providing exact values and partial answers to a question about the limit of these densities as parameters grow large.

Contribution

The authors determine some exact local Turán densities for hypergraphs and show that the previously conjectured equality does not hold in general.

Findings

Calculated exact local Turán densities for specific parameters.

Provided partial answers to an open question about the limit behavior.

Showed that the conjectured equality does not hold universally.

Abstract

For integers $q\ge p\ge r\ge2$, we say that an $r$-uniform hypergraph $H$ has property $(q,p)$, if for any $q$-vertex subset $Q$ of $V(H)$, there exists a $p$-vertex subset $P$ of $Q$ spanning a clique in $H$. Let $T_{r}(n,q,p)=\min\{ e(H): H\subset \binom{[n]}{r}, H \text{~has property~} (q,p)\}$. The local Tur\'an density about property $(q,p)$ in $r$-uniform hypergraphs is defined as $t_{r}(q,p)=\lim_{n\to \infty}T_{r}(n,q,p)/\binom{n}{r}$. Frankl, Huang and R\"odl [J. Comb. Theory, Ser. A, 177 (2021)] showed that $\lim_{p\to\infty}t_{r}(ap+1,p+1)=\frac{1}{a^{r-1}}$ for positive integer $a$ and $t_{3}(2p+1,p+1)=\frac{1}{4}$ for all $p\ge 3$ and asked the question that determining the value of $\lim_{p\to\infty}t_{r}(\gamma p+1,p+1)$, where $\gamma\ge 1$ is a real number. Based on the study of hypergraph Tur\'an densities, we determine some exact values of local Tur\'an densities and answer their question partially; in particular, our results imply that the equality in their question about exact values does not hold in general.

Equations

$$\mathrm{ex}(n,\mathcal{F})=\left(\frac{p-1}{2p}+o(1)\right)n^2,$$

$$\pi (K_k^3)=1-{\left(\frac{2}{k-1}\right)}^2.$$

$$t_r(q,p)=1-\pi ({\mathcal{G}}_{q,p}^r).$$

$$\underset{p\to \infty }{\lim }{t}_r(ap+1,p+1)=\frac{1}{a^{r-1}}.$$

$$t_3(2p+1,p+1)=\frac{1}{4}.$$

$$\underset{p\to \infty }{\lim }{t}_r(\gamma p+1,p+1)=1-\underset{F\in \mathcal{F}}{\min }\pi (F)=\frac{1}{\lfloor \gamma {\rfloor }^{r-1}},$$

$$\underset{p\to \infty }{\lim }{t}_r(\lfloor \gamma p\rfloor +1,p+1)=1-\pi ({\mathcal{F}}_{\gamma }^r),$$

$$t_3(\lfloor \gamma p\rfloor +1,p+1)=\{\begin{array}{ll}\frac{7}{9},& \text{ if }\frac{3}{2}\le \gamma <\frac{5}{3},\\ \frac{19}{27},& \text{ if }\frac{5}{3}\le \gamma <\frac{7}{4},\\ \frac{1}{4},& \text{ if }2\le \gamma <\frac{7}{3},\\ \frac{1}{9},& \text{ if }3\le \gamma <\frac{22}{7}.\end{array}$$

$$e(H[Y])\ge (1-ϵ)(1-\pi ({\mathcal{F}}_{\gamma }^r))\left(\genfrac{}{}{0ex}{}{\ell }{r}\right).$$

$$e(H[Y])=\left(\genfrac{}{}{0ex}{}{\ell }{r}\right)-e(\overline{H[Y]})\ge \left(\genfrac{}{}{0ex}{}{\ell }{r}\right)-\mathrm{ex}(\ell ,{\mathcal{F}}_{\gamma }^r)\ge (1-ϵ)(1-\pi ({\mathcal{F}}_{\gamma }^r))\left(\genfrac{}{}{0ex}{}{\ell }{r}\right).$$

$${\theta }_0=\min \{w-\gamma v:w>\gamma v,w\le \ell ,v\ge r-1\}.$$

$$X_{i+1}=X_i\setminus Z_i,H_{i+1}=H_i[X_{i+1}].$$

$$e(q_i-w_i,p_i-v_i)=\gamma (p_i-v_i)-(q_i-w_i)=\left(\gamma p_i-q_i\right)-\left(\gamma v_i-w_i\right)\ge e(q_i,p_i)+{\theta }_0.$$

$$\gamma (r-1)\le \gamma v_i<|X_i|-|X_{i-1}|=w_i\le \ell .$$

$$\frac{e(H_i)}{\left(\genfrac{}{}{0ex}{}{|X_i|}{r}\right)}=\frac{\mathbb{E}[e(H_i[Q])]}{\left(\genfrac{}{}{0ex}{}{\ell }{r}\right)}\ge (1-\frac{ϵ}{2})(1-\pi ({\mathcal{F}}_{\gamma }^r)).$$

$$e(H)\geq e(H_{i})\geq\big{(}1-\frac{\epsilon}{2}\big{)}^{2}(1-\pi(\mathcal{F}^{r}_{\gamma}))\binom{n}{r}\geq(1-\epsilon)(1-\pi(\mathcal{F}^{r}_{\gamma}))\binom{n}{r}.$$

$$\lfloor \gamma L\rfloor =q_0=q_m+\sum _{i=0}^{m-1}{w}_i\le 2\ell +m\ell <2\ell +\frac{2(\gamma -1){\ell }^2}{{\theta }_0}<\frac{4\gamma {\ell }^2}{{\theta }_0},$$

$$t_r(\lfloor (a+1-\frac{1}{k})p\rfloor +1,p+1)\le {\rho }_r(a,k).$$

$$e(H)$$

$$=\sum _{i=1}^k(-1{)}^{i+1}\sum _{0\le {\ell }_1<{\ell }_2<\cdots <{\ell }_{k+1-i}\le k}e(H[U_{{\ell }_1}\cup U_{{\ell }_2}\cup \cdots \cup U_{{\ell }_{k+1-i}}])$$

$$=\sum _{i=1}^k(-1{)}^{i+1}(\genfrac{}{}{0ex}{}{k+1}{k+1-i}\left)\right(\genfrac{}{}{0ex}{}{\frac{k+1-i}{k+1}n}{r}).$$

$$\frac{{\sum }_{i=1}^k(-1{)}^{i+1}(\genfrac{}{}{0ex}{}{k+1}{k+1-i}\left)\right(\genfrac{}{}{0ex}{}{\frac{k+1-i}{k+1}n}{r})}{\left(\genfrac{}{}{0ex}{}{n}{r}\right)}=\sum _{i=1}^k(-1{)}^{i+1}(\genfrac{}{}{0ex}{}{k+1}{k+1-i}){\left(\frac{k+1-i}{k+1}\right)}^r+o(1),\text{ as }n\to \infty .$$

$$t_r(\lfloor (a+\frac{1}{k})p\rfloor +1,p+1)\le {\eta }_r(a,k).$$

$${\mu }^1$$

$${\mu }^2$$

$${\mu }^3$$

$${\mu }^4$$

$${\mathcal{H}}_a=\{H:(\genfrac{}{}{0ex}{}{P}{3})\subset E(H)\subset H_a,\text{ for all }y,z\in Q,d_H(yz)\ge 2a-1\}.$$

$$d_G(i)$$

$$\le 2a^2+2an-4a^2+(1-\frac{1}{a^2})\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)$$

$$\le (1-\frac{1}{a^2}+\epsilon )\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)+a(2n-1).$$

$$e(G-i)=e(G)-d_G(i)\ge (1-\frac{1}{a^2}+\epsilon )\left(\genfrac{}{}{0ex}{}{n-1}{3}\right)+a(n-1{)}^2+C>\mathrm{ex}(n-1,{\mathcal{H}}_a).$$

$$\mathrm{ex}(n,{\mathcal{H}}_a)<(1-\frac{1}{a^2}+\epsilon )\left(\genfrac{}{}{0ex}{}{n}{3}\right)+a{n}^2+C.$$

$$t_3(\lfloor \gamma p\rfloor +1,p+1)=\frac{1}{a^2}.$$

Full text

1 Introduction

A hypergraph $H$ is a pair $(V,E)$, where $V$ is a finite set of elements, called vertices, and $E$ is a set of nonempty subsets of $V$, called hyperedges. The hypergraph $H$ is said to be $r$-uniform if every hyperedge of $H$ is of size $r$. The simple graphs are the $2$-uniform hypergraphs. We denote the numbers of the vertices and the hyperedges of $H$ by $\nu(H)$ and $e(H)$, respectively. Let $\binom{V(H)}{r}$ be the family of all $r$-element subsets of $V(H)$. For an $r$-uniform hypergraph $H$, we use $\overline{H}$ to denote the complement of $H$, that is $V(\overline{H})=V(H)$ and $E(\overline{H})=\binom{V(H)}{r}\setminus E(H)$. For a vertex set $U\subset V(H)$, denote by $H[U]$ the subhypergraph induced by $U$, that is $H[U]=\{e\in E(H):e\subset U\}.$ A set $I$ of vertices in a hypergraph $H$ is independent if $H[I]$ does not contain a hyperedge. The independence number of $H$, denoted by $\alpha(H)$, is the maximum cardinality of an independent set in $H$. A vertex set $X$ in a hypergraph $H$ is a clique if $H[X]$ is a complete $r$-uniform hypergraph. The clique number of $H$ is the maximum cardinality of a clique in $H$. For two vertices $x,y\in V(H)$, we denote $d_{H}(x)=|\{e\in E(H):e\ni x\}|$ and $d_{H}(xy)=|\{e\in E(H):e\supset\{x,y\}\}|$. Throughout this paper, we write $[n]=\{1,2,\ldots,n\}.$

1.1 Turán problems

The study of Turán number is one of the central topics in extremal graph (hypergraph) theory. Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, we say that an $r$-uniform hypergraph $H$ is $\mathcal{F}$-free if $H$ contains no member of $\mathcal{F}$ as a subhypergraph. The Turán number of $\mathcal{F}$, denoted $\mathrm{ex}(n,\mathcal{F})$, is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-uniform hypergraph on $n$ vertices. We let $\pi(\mathcal{F}):=\lim_{n\to\infty}\frac{\mathrm{ex}(n,\mathcal{F})}{\binom{n}{r}}$ and call $\pi(\mathcal{F})$ the Turán density of $\mathcal{F}$. Katona, Nemetz and Simonovits [14] used an averaging argument to show that the Turán density of any family of hypergraphs exists. If $\mathcal{F}=\{F\}$, we denote the Turán number $\mathrm{ex}(n,\{F\})$ by $\mathrm{ex}(n,F)$ and Turán density $\pi(\{F\})$ by $\pi(F)$, respectively.

The classic Turán Theorem [21] says that for $p\geq 2$, the Turán number $\mathrm{ex}(n,K_{p+1})$ is uniquely attained by the complete balanced $p$-partite graph on $n$ vertices. The following celebrated Erdős-Stone-Simonovits Theorem [6, 7] gives a tight estimate for the Turán number of any family of graphs.

Theorem 1.1** (Erdős-Stone-Simonovits, [6, 7]).**

For any family $\mathcal{F}$ of non-empty graphs, we have

[TABLE]

where $p=\Psi(\mathcal{F})=\min\{\chi(F):F\in\mathcal{F}\}-1$ denotes the subchromatic number of $\mathcal{F}$.

Note that Theorem 1.1 implies that $\pi(F)=\frac{\chi(F)-2}{\chi(F)-1}$ for any non-empty graph $F$. Extending Turán’s Theorem to $r$-uniform hypergraphs is one of the most challenging problems in extremal graph theory. Erdős offered a money prize for determining $\pi(K_{k}^{r})$ for at least one pair $r,k$ with $k>r\geq 3$. However, no Turán density $\pi(K_{k}^{r})$ is known for any $k>r\geq 3$ yet.

Conjecture 1.2** (Turán, [21]).**

For every integer $k\geq 4$,

[TABLE]

The most famous case $k=4$ in Conjecture 1.2, which declares that $\pi(K_{4}^{3})=\frac{5}{9}$ has attracted a lot of interest and activity through the years. It is known that many constructions (e.g. see [3, 8, 15]) achieve the conjectured value. To date, Razborov [19] proved the best known upper bound $\pi(K_{4}^{3})\leq 0.561666$ by using flag algebra.

1.2 Local Turán density

We consider local Turán densities which are defined as follows.

Definition 1.3** (Frankl-Huang-Rödl, [11]).**

For integers $q\geq p\geq r\geq 2$, we say that an $r$-uniform hypergraph $H=(V,E)$ has property $(q,p)$, if for every vertex set $Q\subset\binom{V}{q}$, there exists a vertex set $P\subset\binom{Q}{p}$ spanning a clique in $H$, that is, $\binom{Q}{r}\subset H$. Let $T_{r}(n,q,p)=\min\{e(H):H\subset\binom{[n]}{r},\,H\,\text{has property}\,(q,p)\}$. Let $t_{r}(n,q,p)=\frac{T_{r}(n,q,p)}{\binom{n}{r}}$ and $t_{r}(q,p):=\lim_{n\to\infty}t_{r}(n,q,p).$ We call $t_{r}(q,p)$ the local Turán density of property $(q,p)$ in $r$-uniform hypergraphs.

For $q\geq p\geq r\geq 2$, let $\mathcal{G}^{r}_{q,p}$ be the family of $r$-uniform hypergraphs $G$ with $\nu(G)=q$ and $\alpha(G)\leq p-1$. Note that an $r$-uniform hypergraph $H$ has property $(q,p)$ if and only if $\overline{H}$ is $\mathcal{G}^{r}_{q,p}$-free. Then we have $T_{r}(n,q,p)+\mathrm{ex}(n,\mathcal{G}^{r}_{q,p})=\binom{n}{r}$. Thus,

[TABLE]

This also shows that the local Turán density exists. By the definition of local Turán density, for any integers $q>p>r\geq 2$, $t_{r}(q,p+1)\geq t_{r}(q+1,p+1)\geq t_{r}(q,p)\geq t_{r}(q+1,p).$

In [5], it was shown that for $r=2$ (the graph case), $t_{2}(q,p)=1/\left\lfloor\frac{q-1}{p-1}\right\rfloor$. For general $r$, Frankl and Stechkin [13] showed that if $q\leq\frac{r}{r-1}(p-1)$, then $t_{r}(q,p)=1$. Frankl [10] proved that $\lim_{p\to\infty}t_{3}(2p+1,p+1)=\frac{1}{4}$. Frankl, Huang and Rödl [11] generalize it to the $r$-uniform hypergraphs. They proved the following Theorem.

Theorem 1.4** (Frankl-Huang-Rödl, [11]).**

For integers $r\geq 2$ and $a\geq 2$,

[TABLE]

In the same paper, they [11] obtained the exact value of $t_{3}(2p+1,p+1)$ for all $p\geq 3$.

Theorem 1.5** (Frankl-Huang-Rödl, [11]).**

For every integer $p\geq 3$,

[TABLE]

In light of these results, they ask the the following question.

Question 1.6** (Frankl-Huang-Rödl, [11]).**

Is it possibly true that for every positive real number $\gamma>1$,

[TABLE]

where $\mathcal{F}$ is the family of all the $r$-uniform hypergraphs satifying $\nu(F)\geq\gamma\alpha(F)$?

1.3 Our results

In this section we introduce our main results. Our first result Theorem 1.7 gives a weaker version of the first equality of Question 1.6, while the second result Theorem 1.8 is about some exact values of local Turán densities, which implies that the second equality in Question 1.6 for limit values does not hold in general (on the other hand, it does hold for several sub-intervals).

Theorem 1.7**.**

Let $r\geq 3$ be an integer and $\gamma>1$ be a real number. We have

[TABLE]

where $\mathcal{F}^{r}_{\gamma}=\{r\text{-uniform graphs~{}}F:\nu(F)>\gamma\alpha(F)\}$. Furthermore, for any integer $r\geq 3$, there exist a real number $\gamma$ and a positive integer $p_{0}=p_{0}(r,\gamma)$, such that for any integer $p\geq p_{0}$, we have $t_{r}(\lfloor\gamma p\rfloor+1,p+1)=1-\pi(\mathcal{F}^{r}_{\gamma})$.

Remark 1. From Theorem 1.7, after replacing the hypergraph family $\mathcal{F}$ in Question 1.6 by $\mathcal{F}^{r}_{\gamma}$, if we can show $\pi(\mathcal{F}^{r}_{\gamma})=\min_{F\in\mathcal{F}^{r}_{\gamma}}\pi(F)$,111For a general hypergraph family $\mathcal{G}$, it is known that $\pi(\mathcal{G})=\min_{G\in\mathcal{G}}\pi(G)$ does not hold. then the first equality of Question 1.6 holds.

Theorem 1.8**.**

The following results about local Turán density hold.

(1) For a fixed positive real number $\gamma$, there exists a positive integer $p_{0}=p_{0}(\gamma)$, such that for all integers $p\geq p_{0}$, we have the following

[TABLE]

(2) Let $\gamma$ be a positive real number with $\frac{4}{3}\leq\gamma<\frac{7}{5}$, there exists a positive integer $p_{0}=p_{0}(\gamma)$ such that for all integers $p\geq p_{0}$, we have $t_{4}(\lfloor\gamma p\rfloor+1,p+1)=1-\frac{4!}{4^{4}}$.

(3) For integers $r\geq 3$ and $p\geq r^{2}-r-1$, $t_{r}(rp+1,(r-1)p+1)=1-\frac{r!}{r^{r}}$.

Remark 2. We point out that strictly speaking the first equality in Question 1.6 may not be true. For example, let $r=3$ and $\gamma=\frac{3}{2}$. Then the hypergraph family $\mathcal{F}$ in Question 1.6 contains the single edge as its member. Thus $1-\min_{F\in\mathcal{F}}\pi(F)=1$. But Theorem 1.8 shows that $t_{3}(\left\lfloor\frac{3}{2}p\right\rfloor+1,p+1)=\frac{7}{9}$, which implies that the first equality of Question 1.6 fails in this case. In Theorem 1.7, we modify the corresponding definition to be $\nu(F)>\gamma\alpha(F)$.

Remark 3. It can be seen from the proofs that the conclusions of Theorems 1.7 and 1.8 still hold when replacing $\lfloor\gamma p\rfloor$ with $\lceil\gamma p\rceil$.

The rest of the paper is organized as follows. In Section 2, we prove Theoem 1.7. In Section 3, we give some extremal constructions for local Turán densities and prove Theorem 1.8. In Section 4, we conclude with some remarks and nature problems on this topic.

2 Proof of Theorem 1.7

In this section, we will prove Theorem 1.7. The method comes from the proof of Theorem 1.4 in [11]. For the integer pair $(q,p)$ with $q\leq\gamma p$, we let $e(q,p)=\gamma p-q$. Note that since $q\geq p$, we always have $e(q,p)\leq\gamma q-q=(\gamma-1)q$. We also need the following definition and lemmas.

Definition 2.1** (Frankl-Huang-Rödl, [11]).**

For $H\subset\binom{X}{r}$, the vertex set $Z\subset X$ is a $(w,v)$-hole if $|Z|=w>\gamma v$ and the clique number of $H[Z]$ is $v$.

Lemma 2.2** (Frankl-Huang-Rödl, [11]).**

Suppose $H\subset\binom{X}{r}$ has property $(q,p)$ and $Z$ is a $(w,v)$-hole of $H$ with $w<q$, then $H[X\setminus Z]$ has property $(q-w,p-v)$.

Proof.

Take an arbitrary set $U\in\binom{X\setminus Z}{q-w}$, then $U\cup Z\in\binom{X}{q}$. Since $H$ has property $(q,p)$, $H[U\cup Z]$ contains a clique of size $p$. Hence $H[U]$ contains a clique of size $p-v$. Then $H[X\setminus Z]$ has property $(q-w,p-v)$. ∎

Lemma 2.3**.**

For every $r\geq 3$, $\gamma>1$ and $\epsilon>0$, there exists $\ell_{0}=\ell_{0}(r,\epsilon,\gamma)$ such that the following holds for all $\ell\geq\ell_{0}$. Suppose an $r$-graph $H$ on vertex set $X$ has property $(q,p)$ for all pairs $(q,p)$ with $q\leq\ell$, $p=\lceil\frac{q}{\gamma}\rceil$ (In other words, $H$ does not have a $(w,v)$-hole with $\ell\geq w>\gamma v$). Then for all $Y\in\binom{X}{l}$,

[TABLE]

Proof.

Since $\lim_{\ell\to\infty}\frac{\mathrm{ex}(\ell,\mathcal{F}^{r}_{\gamma})}{\binom{\ell}{r}}=\pi(\mathcal{F}^{r}_{\gamma}),$ there exists $\ell_{0}=\ell_{0}(r,\epsilon,\gamma)$ such that for all $\ell\geq\ell_{0}$, $\mathrm{ex}(\ell,\mathcal{F}^{r}_{\gamma})\leq(\pi(\mathcal{F}^{r}_{\gamma})+\epsilon(1-\pi(\mathcal{F}^{r}_{\gamma})))\binom{\ell}{r}$. For $Y\subset\binom{X}{\ell}$, $H[Y]$ has property $(q,p)$ for all pairs $(q,p)$ with $q\leq\ell$, $p=\lceil\frac{q}{\gamma}\rceil$, which means that $\overline{H[Y]}$ is $\mathcal{F}^{r}_{\gamma}$-free. Thus, we have

[TABLE]

∎

Now we are ready to prove Theorem 1.7.

Proof of Theorem 1.7.

Let $H$ be a maximum $\mathcal{F}^{r}_{\gamma}$-free $r$-graph on $n$ vertices, then $\overline{H}$ has property $(\lfloor\gamma p\rfloor+1,p+1)$ (also each of the properties $(\lceil\gamma p\rceil+1,p+1)$, $(\lfloor\gamma p\rfloor,p)$, and $(\lceil\gamma p\rceil,p)$). Thus, $t_{r}(\lfloor\gamma p\rfloor+1,p+1)\leq 1-\pi(\mathcal{F}^{r}_{\gamma}).$

Now we focus on the lower bound. Given $\epsilon>0$, let us fix a large integer $\ell\geq\ell_{0}=\ell_{0}(r,\frac{\epsilon}{2},\gamma)$, where $\ell_{0}(r,\frac{\epsilon}{2},\gamma)$ is obtained from Lemma 2.3. Let

[TABLE]

And then fix a much larger integer $L\geq\frac{4\gamma\ell^{2}}{\theta_{0}}$. Consider a sufficiently large $r$-uniform hypergraph $H\subset\binom{[n]}{r}$ having property $(q,p)$, where $q=\lfloor\gamma L\rfloor$ and $p=L$. Our aim is to find a subset $X\subset[n]$ with $\binom{|X|}{r}>(1-\frac{\epsilon}{2})\binom{n}{r}$ such that $H[X]$ has no $(w,v)$-hole with $w\leq\ell$ and $v\geq r-1$.

We start with $H_{0}=H$ and define $H_{i}$ inductively. Let $q_{0}=q$, $p_{0}=p$ and $X_{0}=[n]$. Suppose that $H_{i}\subset\binom{X_{i}}{r}$ has property $(q_{i},p_{i})$ and it still has a $(w_{i},v_{i})$-hole. Then we take such a $(w_{i},v_{i})$-hole $Z_{i}\subset X_{i}$ and set

[TABLE]

By Lemma 2.2, $H_{i+1}$ has property $(q_{i}-w_{i},p_{i}-v_{i})$. Moreover,

[TABLE]

Set $q_{i+1}=q_{i}-w_{i}$, $p_{i+1}=p_{i}-v_{i}$ and repeat. At every step, we have

[TABLE]

Since $v_{i}\geq r-1$ for all $i$ and $p_{0}=p$, we have $i\leq\frac{p}{r-1}$. Suppose at step $i$, the hypergraph $H_{i}$ no longer contains a $(w,v)$-hole with $w\leq\ell$. Then we choose a subset $Q$ of size $\ell$ of $V(H_{i})$ uniformly at random. By Lemma 2.3, we have

[TABLE]

On the other hand, $|X_{i}|\geq n-i\ell\geq n-\frac{p\ell}{r-1}$. Thus, for sufficiently large $n$, $\binom{|X_{i}|}{r}\geq(1-\frac{\epsilon}{2})\binom{n}{r}$. Therefore,

[TABLE]

Otherwise suppose this process continues to produce $(w,v)$-holes. Let $m$ be the first index such that $q_{m}<2l$. Since $e(q_{m},p_{m})\leq(\gamma-1)q_{m}$ and $e(q_{i},p_{i})$ strictly increases at least $\theta_{0}$ after each step, it follows that $m\leq\frac{(\gamma-1)q_{m}}{\theta_{0}}$. Thus,

[TABLE]

contradicting that $L\geq\frac{4\gamma\ell^{2}}{\theta_{0}}$. We finish the proof of the first part of Theorem 1.7. The proof of the other part is given by Theorem 1.8. ∎

3 Exact values of some local Turán densities

In this section, we determine some exact values of local Turán densities based on the known hypergraph Turán densities.

3.1 Extremal constructions

In this section, we construct two hypergraph families, which provide upper bounds for local Turán densities.

Definition 3.1**.**

For integers $n,a\geq 1,r\geq 3$ and $k\geq 1$, let $\mathcal{K}^{r}_{n,a,k}$ be the family of $r$-uniform hypergraphs $H$ with $\nu(H)=n$ and $V(H)$ admits a vertex partition $V(H)=V_{1}\cup V_{2}\cup\cdots\cup V_{a}$ and $V_{a}=U_{0}\cup U_{1}\cup\cdots\cup U_{k}$, such that $E(H)=\left(\bigcup_{i=1}^{a-1}\binom{V_{i}}{r}\right)\bigcup\binom{V_{a}\setminus U_{0}}{r}\bigcup\left(\bigcup^{k}_{j=1}\binom{U_{0}\cup U_{j}}{r}\right).$ Let $\rho_{r}(n,a,k)=\min\{e(H):H\in\mathcal{K}^{r}_{n,a,k}\}$ and $\rho_{r}(a,k)=\lim_{n\to\infty}\frac{\rho_{r}(n,a,k)}{\binom{n}{r}}$.

When $k=1$, it is the extremal construction considered in [11] and we have $\rho_{r}(a,1)=\frac{1}{a^{r-1}}.$ We compute its exact value for the case $r=3,a=1$ by the following proposition.

Proposition 3.2**.**

Let $k$ be an integer. We have $\rho_{3}(1,k)=\frac{5k+4}{9k}.$

Proof.

Let $H\in\mathcal{K}^{3}_{n,1,k}$. Let $V(H)=U_{0}\cup U_{1}\cup\cdots\cup U_{k}$ and $E(H)=\binom{V(H)}{3}\setminus\bigcup_{1\leq i_{1}<i_{2}\leq k}U_{0}\times U_{i_{1}}\times U_{i_{2}}.$ We may assume that $|U_{1}|=|U_{2}|=\cdots=|U_{k}|=xn$. Then $|U_{0}|=(1-kx)n$ and $0\leq x\leq\frac{1}{k}$. Thus $e(H)=\binom{n}{3}-\binom{k}{2}(xn)^{2}(1-kx)n$. Denote $f(x)=x^{2}(1-kx)$ where $0\leq x\leq\frac{1}{k}$. By direct calculation, we have $f_{\max}=f(\frac{2}{3k})=\frac{4}{27k^{2}}.$ Thus we have $\rho_{3}(1,k)=\lim_{n\to\infty}\frac{\binom{n}{3}-\binom{k}{2}f_{\max}n^{3}}{\binom{n}{3}}=\frac{5k+4}{9k}.$ ∎

Theorem 3.3**.**

Let $a,k\geq 1$, $r\geq 3$ and $p\geq r-1$ be integers. For large $n$ and any $r$-uniform hypergraph $H\in\mathcal{K}^{r}_{n,a,k}$, $H$ has property $(\lfloor(a+1-\frac{1}{k})p\rfloor+1,p+1)$. Furthermore, we have

[TABLE]

Proof.

Let $V(H)=V_{1}\cup V_{2}\cup\cdots\cup V_{a}$ be the vertex set partition of $V(H)$, where $V_{a}=U_{0}\cup U_{1}\cup\cdots\cup U_{k}$. For any vertex set $X\in\binom{V(H)}{\lfloor(a+1-\frac{1}{k})p\rfloor+1}$, we will show that $H[X]$ contains a clique of size $p+1$.

We may assume that $|X\cap V_{i}|\leq p$ for any $1\leq i\leq a-1$, otherwise $H[X\cap V_{i}]$ contains a clique of size at least $p+1$. Thus, $|X\cap V_{a}|\geq 2p+1-\lfloor\frac{p}{k}\rfloor$. If $p<2k$, then $|X\cap V_{a}|\geq 2p$, one can easily find a clique of size at least $p+1$ in $H[X\cap V_{a}]$. Now we assume that $p\geq 2k$. Simiarly, we may assume that $p+1-\lfloor\frac{p}{k}\rfloor\leq|X\cap U_{0}|\leq p$, otherwise either $H[X\cap U_{0}]$ or $H[X\cap(U_{1}\cup\cdots\cup U_{k})]$ contains a clique of size at least $p+1$. Assume now that $|X\cap U_{0}|=p-\lfloor\frac{p}{k}\rfloor+t$, where $1\leq t\leq\lfloor\frac{p}{k}\rfloor$. Then $|X\cap(U_{1}\cup\cdots\cup U_{k})|\geq p+1-t$. By the pigeonhole principle, there is some $j$ with $1\leq j\leq k$ such that $|X\cap U_{j}|\geq\lceil\frac{p+1-t}{k}\rceil$. Thus, $|X\cap(U_{0}\cup U_{j})|\geq p-\lfloor\frac{p}{k}\rfloor+t+\lceil\frac{p+1-t}{k}\rceil\geq p+1$. Therefore $H[X\cap(U_{0}\cup U_{j})]$ contains a clique of size at least $p+1$. ∎

Definition 3.4**.**

For integers $n,a\geq 1,r\geq 3$ and $2\leq k\leq r-1$, let $\mathcal{L}^{r}_{n,a,k}$ be the family of $r$-uniform hypergraphs $H$ with $\nu(H)=n$ and $V(H)$ admits a vertex partition $V(H)=V_{1}\cup V_{2}\cup\cdots\cup V_{a}$ and $V_{a}=U_{0}\cup U_{1}\cup\cdots\cup U_{k},$ such that $E(H)=\left(\bigcup_{i=1}^{a-1}\binom{V_{i}}{r}\right)\bigcup\left(\bigcup^{k}_{j=0}\binom{V_{a}\setminus U_{j}}{r}\right).$ Let $\eta_{r}(n,a,k)=\min\{e(H):H\in\mathcal{L}^{r}_{n,a,k}\}$ and $\eta_{r}(a,k)=\lim_{n\to\infty}\frac{\eta_{r}(n,a,k)}{\binom{n}{r}}.$

We compute its exact value for the case $a=1$ by the following proposition.

Proposition 3.5**.**

Let $r\geq 3$ and $2\leq k\leq r-1$. We have $\eta_{r}(1,k)=\sum_{i=1}^{k}(-1)^{i+1}\binom{k+1}{k+1-i}(\frac{k+1-i}{k+1})^{r}.$

Proof.

Let $H\in\mathcal{L}^{3}_{n,1,k}$ with the minmum number of hyperedges. Let $V(H)=U_{0}\cup U_{1}\cup\cdots\cup U_{k}$ and $E(H)=\bigcup^{k}_{j=0}\binom{V(H)\setminus U_{j}}{r}.$ We may assume that $|U_{0}|$=$|U_{1}|=|U_{2}|=\cdots=|U_{k}|=\frac{n}{k+1}$. By the inclusion-exclusion principle, we have

[TABLE]

Then the edge density of $H$ is

[TABLE]

∎

Remark 3.6**.**

Let $r\geq 3$. We have $\eta_{r}(1,r-1)=1-\frac{r!}{r^{r}}.$

Theorem 3.7**.**

Let $a\geq 1,r\geq 3$, $p\geq r-1$ and $2\leq k\leq r-1$ be integers. For large $n$ and any $r$-uniform hypergraph $H\in\mathcal{L}^{r}_{n,a,k}$, $H$ has property $(\lfloor(a+\frac{1}{k})p\rfloor+1,p+1)$. Furthermore, we have

[TABLE]

Proof.

Let $V(H)=V_{1}\cup V_{2}\cup\cdots\cup V_{a}$ be the vertex set partition of $V(H)$, where $V_{a}=U_{0}\cup U_{1}\cup\cdots\cup U_{k}$. For any vertex set $X\in\binom{V(H)}{\lfloor(a+\frac{1}{k})p\rfloor+1}$, we will show that $H[X]$ contains a clique of size $p+1$.

We may assume that $|X\cap V_{i}|\leq p$ for any $1\leq i\leq a-1$, otherwise $X\cap V_{i}$ contains a clique of size at least $p+1$. Thus, $|X\cap V_{a}|\geq\lfloor\frac{(k+1)p}{k}\rfloor+1\geq\frac{(k+1)p+1}{k}$. By the pigeonhole principle, there is some $j$ with $0\leq j\leq k$ such that $|X\cap(V_{a}\setminus U_{j})|\geq\lceil\frac{k}{k+1}|X\cap V_{a}|\rceil\geq p+1$. Thus $H[X]$ contains a clique of size at least $p+1$. ∎

3.2 Proof of Theorem 1.8

In this subsection, we prove Theorem 1.8. The key tool we use is the “blow-up” operation of the $r$-uniform hypergraphs. Let $H$ be an $r$-uniform hypergraph on vertex set $\{v_{1},v_{2},\ldots,v_{l}\}$ and $\boldsymbol{\mathbf{\mu}}=(\mu_{1},\mu_{2},\ldots,\mu_{l})$ be an integer vector with each $\mu_{i}\geq 1$. Then the blow-up $H(\boldsymbol{\mathbf{\mu}})$ is the $r$-uniform hypergraph formed by replacing the vertex $v_{i}$ of $H$ with a disjoint class of $\mu_{i}$ vertices for each $i$ and inserting a complete $r$-partite $r$-uniform hypergraph between any vertex classes corresponding to an edge in $H$. Given a family $\mathcal{H}=\{H_{1},H_{2},\ldots,H_{s}\}$ of $r$-uniform hypergraphs and a family $\mathcal{T}=\{\boldsymbol{\mathbf{\mu}}^{1},\boldsymbol{\mathbf{\mu}}^{2},\ldots,\boldsymbol{\mathbf{\mu}}^{s}\}$ of positive integer vectors with each $\boldsymbol{\mathbf{\mu}}^{i}$ of dimension $|V(H_{i})|$, we define the $\mathcal{T}$-blow-up of $\mathcal{H}$ to be $\mathcal{H}(\mathcal{T})=\{H_{i}(\boldsymbol{\mathbf{\mu}}^{i}):1\leq i\leq s\}$. Brown and Simonovits [4] proved the following extremely useful result.

Theorem 3.8** (Brown-Simonovits, [4]).**

If $\mathcal{H}=\{H_{1},H_{2},\ldots,H_{s}\}$ is a family of $r$-uniform hypergraphs and $\mathcal{T}=\{\boldsymbol{\mathbf{\mu}}^{1},\boldsymbol{\mathbf{\mu}}^{2},\ldots,\boldsymbol{\mathbf{\mu}}^{s}\}$ is a family of positive integer vectors with each $\boldsymbol{\mathbf{\mu}}^{i}$ of dimension $|V(H_{i})|$, then $\pi(\mathcal{H}(\mathcal{T}))=\pi(\mathcal{H})$.

We also need the following results on hypergraph Turán densities.

Theorem 3.9** (Frankl-Füredi, [9]).**

Let $F_{5}=\{123,124,345\}$, we have $\pi(F_{5})=\frac{2}{9}.$

Theorem 3.10** (Baber-Talbot, [1]).**

Let $H_{1}=\{123,124,134,234\},H_{2}=\{123,124,125,345,346\},\\ H_{3}=\{123,124,345,156,256\}$, $H_{4}=\{123,124,125,346,356,456\},$ then $\pi(\{H_{1},H_{2},H_{3},H_{4}\})=\frac{8}{27}.$

Let $H^{-}_{2}=\{123,124,125,345\}$. Note that $H_{2}$ is a subgraph of some blow-up of $H^{-}_{2}$, thus we have the following corollary.

Corollary 3.11**.**

$\pi(\{H_{1},H^{-}_{2},H_{3},H_{4}\})=\frac{8}{27}.$**

Theorem 3.12** (Mubayi-Rödl, [17]).**

Let $H_{7}=\binom{[4]}{3}\cup\{(a,x,y):a\in[4],x,y\in\{5,6,7\},x\neq y\}\setminus\{(1,5,6)\}$, then $\pi(H_{7})=\frac{3}{4}.$

For $r\geq 3$, let the generalized triangle $T^{r}$ be the $r$-uniform hypergraph on $(2r-1)$ vertices with three edges $\{1,\ldots,r\},\{1,\ldots,r-1,r+1\}\,\text{and}\,\{r,r+1,\ldots,2r-1\}$. Pikhurko [18] obtained the Turán density of $T^{4}$.

Theorem 3.13** (Pikhurko, [18]).**

$\pi(T^{4})=\frac{4!}{4^{4}}.$**

For $r\geq 3$, the generalized fan denoted by $F^{r}$ is the $r$-uniform hypergraph comprising $r+1$ edges $e_{1},\ldots,e_{r},e$ such that $e_{i}\cap e_{j}=\{x\}$ for all $i\neq j$, where $x\notin e$ and $|e_{i}\cap e|=1$ for all $i$. Mubayi and Pikhurko [16] obtained the Turán density of $F^{r}$.

Theorem 3.14** (Mubayi-Pikhurko, [16]).**

$\pi(F^{r})=\frac{r!}{r^{r}}.$**

Now we are ready to prove Theorem 1.8.

Proof of Theorem 1.8.

(1) If $\frac{3}{2}\leq\gamma<\frac{5}{3}$, by Theorem 3.3 and Proposition 3.2, $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\leq t_{3}(\lfloor\frac{3}{2}p\rfloor+1,p+1)\leq\rho_{3}(1,2)=\frac{7}{9}.$ We just need to prove the lower bound. Let $\boldsymbol{\mathbf{\mu}}=(\frac{\lfloor\gamma p\rfloor+1}{5},\frac{\lfloor\gamma p\rfloor+1}{5},\frac{\lfloor\gamma p\rfloor+1}{5},\frac{\lfloor\gamma p\rfloor+1}{5},\frac{\lfloor\gamma p\rfloor+1}{5})$222When $\frac{\lfloor\gamma p\rfloor+1}{5}$ is not an integer, we always let it be its floor or ceil such that the sum of all the elements in $\boldsymbol{\mathbf{\mu}}$ equals $\lfloor\gamma p\rfloor+1$. We do the same if similar situations occur in the rest of the paper.. By Theorem 3.8 and Theorem 3.9, we have $\pi(F_{5}(\boldsymbol{\mathbf{\mu}}))=\pi(F_{5})=\frac{2}{9}$. Also, $\alpha(F_{5}(\boldsymbol{\mathbf{\mu}}))\leq\frac{3(\lfloor\gamma p\rfloor+1)}{5}+O(1)<p+1$ for large $p$. By (1), for large $p$, we have $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\geq 1-\pi(F_{5}(\boldsymbol{\mathbf{\mu}}))=\frac{7}{9}$.

If $\frac{5}{3}\leq\gamma<\frac{7}{4}$, by Theorem 3.3 and Proposition 3.2, $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\leq t_{3}(\lfloor\frac{5}{3}p\rfloor+1,p+1)\leq\rho_{3}(1,3)=\frac{19}{27}$. We just need to prove the lower bound. Let

[TABLE]

Then for large $p$, $\alpha(H_{1}(\boldsymbol{\mathbf{\mu}}^{1}))\leq\frac{\lfloor\gamma p\rfloor+1}{2}+O(1)<p+1$, $\alpha(H^{-}_{2}(\boldsymbol{\mathbf{\mu}}^{2}))\leq\frac{4(\lfloor\gamma p\rfloor+1)}{7}+O(1)<p+1$, $\alpha(H_{3}(\boldsymbol{\mathbf{\mu}}^{3}))\leq\frac{4(\lfloor\gamma p\rfloor+1)}{7}+O(1)<p+1$, $\alpha(H_{4}(\boldsymbol{\mathbf{\mu}}^{4}))\leq\frac{4(\lfloor\gamma p\rfloor+1)}{7}+O(1)<p+1$. By (1), Theorem 3.8 and Corollary 3.11, for large $p$, we have $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\geq 1-\pi(\{H_{1}(\boldsymbol{\mathbf{\mu}}^{1}),H^{-}_{2}(\boldsymbol{\mathbf{\mu}}^{2}),H_{3}(\boldsymbol{\mathbf{\mu}}^{3}),H_{4}(\boldsymbol{\mathbf{\mu}}^{4})\})=1-\frac{8}{27}=\frac{19}{27}$.

If $2\leq\gamma<\frac{7}{3}$, by Theorem 3.3 and Proposition 3.2, $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\leq t_{3}(2p+1,p+1)\leq\rho_{3}(2,1)=\frac{1}{4}$. We just need to prove the lower bound. Let $\boldsymbol{\mathbf{\mu}}=(\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7})$. By Theorem 3.8 and Theorem 3.12, we have $\pi(H_{7}(\boldsymbol{\mathbf{\mu}}))=\pi(H_{7})=\frac{3}{4}.$ Also, $\alpha(H_{7}(\boldsymbol{\mathbf{\mu}}))\leq\frac{3(\lfloor\gamma p\rfloor+1)}{7}+O(1)<p+1$ for large $p$. By (1), for large $p$, we have $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\geq 1-\pi(H_{7}(\boldsymbol{\mathbf{\mu}}))=\frac{1}{4}$.

For $3\leq\gamma<\frac{22}{7}$, the proof will be given in subsection 3.3 (See Theorem 3.20).

(2) For $\frac{4}{3}\leq\gamma<\frac{7}{5}$, by Theorem 3.7 and Remark 3.6, $t_{4}(\lfloor\gamma p\rfloor+1,p+1)\leq t_{4}(\lfloor\frac{4}{3}p\rfloor+1,p+1)\leq\eta_{4}(1,3)=1-\frac{4!}{4^{4}}$. Now we focus on proving the lower bound. Let $\boldsymbol{\mathbf{\mu}}=(\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7},\frac{\lfloor\gamma p\rfloor+1}{7})$. By Theorem 3.8 and Theorem 3.13, we have $\pi(T^{4}(\boldsymbol{\mathbf{\mu}}))=\pi(T^{4})=\frac{4!}{4^{4}}$. Also, $\alpha(T^{4}(\boldsymbol{\mathbf{\mu}}))\leq\frac{5(\lfloor\gamma p\rfloor+1)}{7}+O(1)<p+1$ for large $p$. By (1), for large $p$, we have $t_{4}(\lfloor\gamma p\rfloor+1,p+1)\geq 1-\pi(T^{4}(\boldsymbol{\mathbf{\mu}}))=1-\frac{4!}{4^{4}}$.

(3) By Theorem 3.7 and Remark 3.6, $t_{r}(rp+1,(r-1)p+1)\leq\eta_{r}(1,r-1)=1-\frac{r!}{r^{r}}$. Now we focus on proving the lower bound. For $r\geq 3$ and $p\geq r^{2}-r-1$, let $F^{r}_{p}$ be the blow-up of $F^{r}$ by replacing each vertex of $e$ with $p-r^{2}+2r$ vertices, the common vertex $x$ with $(r-1)^{2}$ vertices and each vertex of $V(F^{r})\setminus(e\cup\{x\})$ with $r-1$ vertices. By Theorem 3.8 and Theorem 3.14, we have $\pi(F^{r}_{p})=\pi(F^{r})=\frac{r!}{r^{r}}$. Note that $\nu(F^{r}_{p})=rp+1$. Also, $\alpha(F^{r}_{p})=(r-1)p$. By (1), we have $t_{r}(rp+1,(r-1)p+1)\geq 1-\pi(F_{p}^{r})+O(1)=1-\frac{r!}{r^{r}}$. ∎

3.3 Proof of Theorem 1.8 when $3\leq\gamma<\frac{22}{7}$

To obtain more exact values of local Turán densities, we consider an extension of Theorem 3.12. We define the family $\mathcal{H}_{a}$ of $3$-uniform hypergraphs.

Definition 3.15**.**

For an integer $a\geq 2$, let $P$ and $Q$ be two disjoint vertex sets with $|P|=2a$ and $Q=a+1$. Let $H_{a}$ be the $3$-uniform hypergraph on vertex set $P\cup Q$ with $E(H_{a})=\binom{P}{3}\cup\{xyz:x\in P,y,z\in Q\}$. Let

[TABLE]

Using Turán’s Theorem [21], we have the following lemma about Turán number in multigraphs.

Lemma 3.16**.**

Let $n\geq a\geq 2$ be integers and $G$ be a multigraph on $n$ vertices. If any $a+1$ vertices of $G$ will induce at least one edge with multiplicities at least $2$, then $e(G)\geq\frac{2}{a}\binom{n}{2}-n.$

Proof.

By Turán’s Theorem [21], $\mathrm{ex}(n,K_{a+1})\leq(1-\frac{1}{a})\binom{n}{2}+\frac{n}{2}$. Thus $T_{2}(n,a+1,2)=\binom{n}{2}-\mathrm{ex}(n,K_{a+1})\geq\frac{1}{a}\binom{n}{2}-\frac{n}{2}$. Hence $e(G)\geq 2\times T_{2}(n,a+1,2)\geq\frac{2}{a}\binom{n}{2}-n.$ ∎

Now we will compute the Turán density of family $\mathcal{H}_{a}$ under the condition that a warm version of Turán’s Conjecture (Conjecture 1.2) holds.

Theorem 3.17**.**

Let $a\geq 2$ be an integer. If $\pi(K^{3}_{2a})\leq 1-\frac{1}{a^{2}}$, then $\pi(\mathcal{H}_{a})=1-\frac{1}{a^{2}}$.

Proof.

Let $G_{n,a}=\binom{[n]}{3}\setminus(\binom{V_{1}}{3}\cup\binom{V_{2}}{3}\cup\cdots\cup\binom{V_{a}}{3})$ with $V_{1}\sqcup V_{2}\sqcup\cdots\sqcup V_{a}=[n]$ and $V_{i}\in\{\lfloor\frac{n}{a}\rfloor,\lceil\frac{n}{a}\rceil\}.$ It is easy to check that $G_{n,a}$ is $\mathcal{H}_{a}$-free and $e(G_{n,a})=(1-\frac{1}{a^{2}}+o(1))\binom{n}{3}$ as $n\to\infty$. Thus $\pi(\mathcal{H}_{a})\geq 1-\frac{1}{a^{2}}$.

For the upper bounds, we just need to show that for any real number $0<\varepsilon<\frac{1}{a^{2}}$, there exist an integer $N=N(\varepsilon)$ and a real number $C=C(N)$, such that for any integer $n\geq N$, $\mathrm{ex}(n,\mathcal{H}_{a})<(1-\frac{1}{a^{2}}+\varepsilon)\binom{n}{3}+an^{2}+C$. We prove it by induction on $n\geq N$. Since $\pi(K^{3}_{2a})\leq 1-\frac{1}{a^{2}}$, we can choose a large integer $N_{1}=N_{1}(\varepsilon)$ such that $\mathrm{ex}(n,K^{3}_{2a})\leq(1-\frac{1}{a^{2}}+\varepsilon)\binom{n}{3}$ holds for $n\geq N_{1}$. Furthermore, we can choose an integer $N=N(\varepsilon)\geq N_{1}$ and a real number $C$ such that $(1-\frac{1}{a^{2}}+\varepsilon)\binom{N}{3}+aN^{2}+C=\binom{N}{3}$ and $(1-\frac{1}{a^{2}}+\varepsilon)\binom{n}{3}+an^{2}+C<\binom{n}{3}$ holds for $n>N$.

It is easy to see that our claim holds for $n=N$. Now let $n>N$ and assume that our claim holds for $n-1$. Let $G$ be a $3$-uniform hypergraph on vertex set $[n]$ with $e(G)\geq(1-\frac{1}{a^{2}}+\varepsilon)\binom{n}{3}+an^{2}+C$. Therefore, $G$ contains a clique with $2a$ vertices. By symmetry, suppose $\binom{[2a]}{3}\subset G$. For $i\in[2a]$, we define the link graphs $G(i)=\left\{\{x,y\}\in\binom{[2a+1,n]}{2}:ixy\in G\right\}$. Let $M$ be the multigraph whose edge set is the union (with multiplicities) $G(1)\cup G(2)\cup\cdots\cup G(2a)$. Let $\overline{M}=\overline{G(1)}\cup\overline{G(2)}\cup\cdots\cup\overline{G(2a)}$. If there exists an $(a+1)$-subset $S\subset[2a+1,n]$, such that the multiplicity of each edge of the induced subgraph of $\overline{M}$ on $S$ is no more than $1$, in other words, the multiplicity of each edge of the induced subgraph of $M$ on $S$ is at least $2a-1$, then vertex set $[1,2a]\cup S$ induces some copy of hypergraph in $\mathcal{H}_{a}$ in $G$. Thus, we assume that the induced subgraph of $\overline{M}$ on any $(a+1)$-subset of $[2a+1,n]$ will contains at least one edge with multiplicities at least $2$. By Lemma 3.16, $e(\overline{M})\geq\frac{2}{a}\binom{n-2a}{2}-(n-2a)$. Thus, $e(M)\leq(2a-\frac{2}{a})\binom{n-2a}{2}+(n-2a)$. Therefore, there exists a vertex $i\in[2a]$, such that $e(G(i))\leq\frac{1}{2a}e(M)\leq(1-\frac{1}{a^{2}})\binom{n-2a}{2}+\frac{n}{2a}-1$. It follows that

[TABLE]

By induction hypothesis, we have

[TABLE]

So $G-i$ contains a copy of some hypergraph in $\mathcal{H}_{a}$. It follows that

[TABLE]

∎

Theorem 3.18**.**

Let $a\geq 2$ be an integer. If $\pi(\mathcal{H}_{a})=1-\frac{1}{a^{2}}$, then for any real number $\gamma$ with $a\leq\gamma<a+\frac{1}{2a+1}$, there exist a postive integer $p_{0}=p(\gamma)$, such that for all integers $p\geq p_{0}$,

[TABLE]

Proof.

By Theorem 3.3, $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\leq t_{3}(ap+1,p+1)\leq\rho_{3}(a,1)=\frac{1}{a^{2}}$. We just need to prove the lower bound. For each hypergraph $H\in\mathcal{H}_{a}$, let $H^{\lfloor\gamma p\rfloor+1}$ be the blow-up of $H$ by replacing each vertex of $P$ with $\frac{a(\lfloor\gamma p\rfloor+1)}{2a^{2}+a+1}$ vertices and replacing each vertex of $Q$ with $\frac{\lfloor\gamma p\rfloor+1}{2a^{2}+a+1}$ vertices. Let $\mathcal{H}^{\lfloor\gamma p\rfloor+1}_{a}=\{H^{\lfloor\gamma p\rfloor+1}:H\in\mathcal{H}_{a}\}$. By Theorem 3.8, $\pi(\mathcal{H}^{\lfloor\gamma p\rfloor+1}_{a})=\pi(\mathcal{H}_{a})=1-\frac{1}{a^{2}}$. For large $p$, $\alpha(H^{\lfloor\gamma p\rfloor+1})\leq\frac{(2a+1)(\lfloor\gamma p\rfloor+1)}{2a^{2}+a+1}+O(1)<p+1$. By (1), $t_{3}(\lfloor\gamma p\rfloor+1,p+1)\geq 1-\pi(\mathcal{H}^{\lfloor\gamma p\rfloor+1}_{a})=\frac{1}{a^{2}}.$ ∎

Vaughan [22] computed an upper bound of $\pi(K_{6}^{3})$ by flag algebra.

Theorem 3.19** (Vaughan, [22], [2]).**

$\pi(K_{6}^{3})\leq 0.8583903.$**

Combining Theorem 3.17, Theorem 3.18 and Theorem 3.19, we have the following conclusion.

Theorem 3.20**.**

Let $\gamma$ be a real number with $3\leq\gamma<\frac{22}{7}$, there exist a postive integer $p_{0}=p(\gamma)$, such that for all integers $p\geq p_{0}$,

[TABLE]

Remark 3.21**.**

Combining Theorem 3.17 and Theorem 3.18, if $\pi(K^{3}_{2a})\leq 1-\frac{1}{a^{2}}$, then for any real number $\gamma$ with $a\leq\gamma<a+\frac{1}{2a+1}$, there exist a postive integer $p_{0}=p(\gamma)$, such that for all integers $p\geq p_{0}$, $t_{3}(\lfloor\gamma p\rfloor+1,p+1)=\frac{1}{a^{2}}.$ If Turán’s Conjecture (Conjecture 1.2) holds for odd number $k=2a+1$, that is $\pi(K^{3}_{2a+1})=1-\frac{1}{a^{2}}$, then for any real number $\gamma$ with $a\leq\gamma<a+\frac{1}{2}$, there exist a postive integer $p_{0}=p(\gamma)$, such that for all integers $p\geq p_{0}$, $t_{3}(\lfloor\gamma p\rfloor+1,p+1)=\frac{1}{a^{2}}.$

4 Concluding remarks

In this paper, we determine some exact values of local Turán densities and answer Question 1.6 partially; in particular, our results imply that the equality in Question 1.6 about exact values of the limits does not hold in general. The local Turán density problems are still widely open. Here we discuss some natural problems.

Frankl, Huang and Rödl [11] extended Theorem 1.5 to $r$-uniform hypergraphs and made the following conjecture.

Conjecture 4.1** (Frankl-Huang-Rödl, [11]).**

For integers $r\geq 2$ and $p$ sufficiently large,

[TABLE]

Note that Question 1.6 is a generalization of Theorem 1.4 and closely related to Conjecture 4.1. Based on Theorem 1.7, we extend Question 1.6 to the following more general question. Recall that, for an integer $r\geq 3$ and a real number $\gamma>1$, we defined the family $\mathcal{F}^{r}_{\gamma}$ of $r$-uniform hypergraphs as $\mathcal{F}^{r}_{\gamma}=\{r\text{-uniform graphs~{}}F:\nu(F)>\gamma\alpha(F)\}.$

Question 4.2**.**

Let $r\geq 3$ be an integer and $\gamma>1$ be a real number. Does there exist a positive integer $p_{0}=p_{0}(r,\gamma)$, such that for all integers $p\geq p_{0}$,

[TABLE]

Using the blow-up theorem of Brown and Simonovits [4], one can show that Question 4.2 is equivalent to the following.

Question 4.3**.**

Let $r\geq 3$ be an integer and $\gamma>1$ be a real number. Does there exist a finite subfamily $\mathcal{F}_{fin}\subset\mathcal{F}^{r}_{\gamma}$ such that

[TABLE]

We say a family $\mathcal{H}$ of $r$-uniform hypergraphs compact if there is a finite subfamily $\mathcal{H}_{fin}\subset\mathcal{H}$ such that $\pi(\mathcal{H}_{fin})=\pi(\mathcal{H})$ (See Conjecture 1 in [4]). In this notion, Question 4.3 asks whether the family $\mathcal{F}^{r}_{\gamma}$ is compact. The result of Frankl and Stechkin [13] shows that the family $\mathcal{F}^{r}_{\gamma}$ is compact when $1\leq\gamma<\frac{r}{r-1}$. Theorem 1.5 shows that the family $\mathcal{F}^{3}_{2}$ is compact. Moreover, the corresponding hypergraph family of each case considered in Theorem 1.8 is also compact.

Acknowledgment. Very recently we learn that Frankl and Nie [12] have also obtained results in this topic, which partially overlap with our Theorem 1.8 when $r=3$ and $2\leq\gamma<7/3$.